WASHINGTON — President Trump has repeatedly warned that terrorists are pouring into the United States from Mexico, in one of his central justifications for building a border wall.

But his own government’s assessments conclude that Mr. Trump has seriously overstated the threat. And counterterrorism officials and experts said there had never been a case of a known terrorist sneaking into the country through open areas of the southwest border.

Despite the administration’s focus on security threats at the border, a White House strategy document sent to Congress last month outlining steps needed to monitor and intercept terrorists included no reference to the need for construction of barriers, fences or walls. Separately, an intelligence analysis concluded that cyberattacks are the top threat to the United States — not terrorists at the border.

“There is no wave of terrorist operatives waiting to cross overland into the United States,” Nicholas J. Rasmussen, the former director of the National Counterterrorism Center, said on Tuesday. “It simply isn’t true.”

In a rare prime-time address to the nation, broadcast at 9 p.m. on Tuesday, Mr. Trump was expected to describe a national security crisis brewing at the Mexican border. It was his opportunity to directly defend his demands for $ 5.7 billion for a border wall — funding that congressional Democrats have refused to provide, fueling the 18-day government shutdown.

Many Latin American countries have border law enforcement gaps — limited law enforcement capabilities and established smuggling routes — that extremists could exploit to harm the United States, according to the State Department’s latest Country Reports on Terrorism.

But, the report concluded, that has not happened.

“These vulnerabilities offer opportunities to foreign terrorist groups, but there have been no cases of terrorist groups exploiting these gaps to move operations through the region,” the report said.

The latest ranking of urgent national security vulnerabilities, compiled annually by American intelligence agencies, puts the terrorist threat from the southwest border low on the list and then mentions it in a discussion of how “worldwide production of cocaine, heroin and methamphetamine is at record levels.”

Over the last several days, White House and Department of Homeland Security officials have relentlessly pushed the case that the situation at the border is both a national security crisis and a humanitarian one.

Sarah Huckabee Sanders, the White House press secretary, said on Sunday that “nearly 4,000 known or suspected terrorists come into our country illegally,” and added that the “most vulnerable point of entry is at our southern border.”

A day later, Kellyanne Conway, the White House counselor, called those comments “an unfortunate misstatement.”

At the same time, in a memo sent to journalists late Monday, Kirstjen Nielsen, the homeland security secretary, cited “more than 3,000 ‘special-interest aliens’ — individuals with suspicious travel patterns” who posed a potential national security risk.

“The threat is real,” she wrote in a series of tweets on Monday afternoon. “The number of terror-watchlisted encountered at our Southern Border has increased over the last two years. The exact number is sensitive and details about these cases are extremely sensitive.”

Former national security officials and analysts have pushed back — especially on the notion that terrorism suspects or their sympathizers use the southwest border as a door to the United States.

“That 4,000 number was bull,” said W. Ralph Basham, who served as commissioner of United States Customs and Border Protection from 2006 to 2009, during the Bush administration.

“The idea that you have that many terrorists flooding across the border when you have all of these dedicated agents focused on stopping that kind of activity is ridiculous,” Mr. Basham said.

A senior American counterterrorism official said on Tuesday night that an annual average of three known or suspected terrorists — all of whom were on watch lists — had tried to enter the United States at a legal border crossing or entry point over the last several years, and had been denied.

A 22-page National Strategy to Combat Terrorism Travel, overseen by the staff of the National Security Council and under Mr. Trump’s signature, cited a broad commitment to making it “more difficult for terrorists to cross U.S. borders.”

Sent to Congress on Dec. 21, the day before the government shutdown began, the document urged data sharing and improved coordination with foreign partners to identify terrorists before they travel. It did not urge the construction of a border wall.

The only explicit reference to “border security” in the document, which was obtained on Tuesday by The New York Times, is a listing of executive orders that Mr. Trump has already signed. It also included strategies to “interdict” potential terrorists with plans to “maintain the equipment and technology” necessary to stop bad actors and share real-time information between law enforcement agencies.

The 2018 assessment by the Office of the Director of National Intelligence ranked cyberattacks against utilities, communications systems and markets as the top threat to the United States. Published annually by American intelligence agencies, the ranking has not changed much over the past five years.

But the government shutdown has furloughed nearly half of the work force of the Department of Homeland Security’s new Cybersecurity and Infrastructure Security Agency, which is advertised as the first line of defense against network attacks. Mr. Trump has also dismantled the office of the White House cybersecurity coordinator, the job that was supposed to sew together offense and defense against the daily barrage of cyberattacks.

After cyberattacks, the top threats to the United States include the rise of smaller, more deadly nuclear and biological weapons; terrorist attacks by the Islamic State and Al Qaeda; and Russian-style influence campaigns and threats to the United States’ space assets.

Counterterrorism officials have long discounted the threat of Islamic State or Qaeda terrorists entering through the southwest border. In some cases, the opposite has happened, said Seamus Hughes, the deputy director of George Washington University’s program on extremism, citing homegrown American extremists who crossed into Mexico to avoid being detected on no-fly lists.

Officials included the case of Jason Ludke of Milwaukee, who pleaded guilty in October to conspiring to provide material support to the Islamic State. He and a co-conspirator were headed to the border in Texas, with plans to join the Islamic State in Syria or Iraq, when they were arrested.

According to the plea agreement, Mr. Ludke had planned to work under the Islamic State’s direction and control and recorded a video of himself pledging his allegiance to the terrorist group’s leader, Abu Bakr al-Baghdadi. Mr. Ludke told an undercover F.B.I. agent that he had training in jujitsu and computers, which he believed would benefit the Islamic State.

Even as Mr. Trump has lobbied for the wall, he has also pressed the Pentagon and the Department of Homeland Security to immediately tighten security on the border.

On Dec. 27, homeland security officials asked for assistance that the Pentagon later estimated would require 3,000 additional military personnel to be sent to the border, according to memos exchanged by the two departments over the last two weeks. The force could include a combination of National Guard troops, active-duty military personnel and volunteers, but that was not made clear in the documents, officials said. It remains an issue of hot debate inside the Pentagon.

The Homeland Security Department also requested 146 mobile surveillance vehicles to be sent to four border states, along with enough personnel and material to install 150 additional miles of barbed concertina wire. The homeland security memo also again asked Defense Department officials to provide additional aviation support to monitor and control the flow of immigrants.

The Pentagon has not yet approved the request. It included guidelines for Ms. Nielsen and other homeland security officials to “take the lead” on strategic communications intended to highlight the Pentagon’s role. It also made clear the troops would carry no weapons.